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Chapter 9: Problem 45

Find a recursive definition for the sequence. $$1,3,6,10,15, \ldots$$

Short Answer

Expert verified
Define the sequence recursively with \(a_1 = 1\) and \(a_n = a_{n-1} + n\).

Step by step solution

01

Identify the Sequence Pattern

Observe the given sequence: \(1, 3, 6, 10, 15, \ldots\). Notice that each term adds an increasing integer to the previous term. Specifically, the differences between consecutive terms are \(2, 3, 4,\) and so on, indicating that each term adds the next integer value.
02

Establish the Recursive Relationship

The sequence starts with the first term \(a_1 = 1\). Each subsequent term is generated by adding an integer, starting from \(2\), to the previous term. This indicates that \(a_n = a_{n-1} + n\).
03

Define the Base Case and Recursive Formula

Specify the base case for the sequence: \(a_1 = 1\). The recursive formula, based on previous observations, is \(a_n = a_{n-1} + n\). This establishes how each term is built from its predecessor and an increasing integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recursive Definition
A recursive definition provides a way to define a sequence of numbers using the terms that come before them. In the context of sequences, each term is calculated by applying a specific rule to the term before it. It creates a chain of terms that depend on one another. For our sequence: \(1, 3, 6, 10, 15, \ldots\), we can observe that it starts with a base case, and each subsequent term is derived using a fixed rule.

  • Base Case: This is the starting point of the sequence. For this sequence, \(a_1 = 1\).
  • Recursive Step: This step defines how to get from one term to the next. For this sequence, each term \(a_n\) is obtained by adding \(n\) to the previous term \(a_{n-1}\). Hence, the recursive formula is \(a_n = a_{n-1} + n\).
By understanding the base case and recursive formula, students can calculate any term in the sequence as long as they know the preceding one.
Sequence Pattern
Sequence patterns reveal the rule or rules that govern how a sequence progresses. Identifying these patterns is crucial in forming recursive definitions with confidence. In the sequence \(1, 3, 6, 10, 15, \ldots\):

  • The differences between consecutive terms are sequential integers: \(2, 3, 4, 5, \ldots\).
  • This indicates a pattern where each term after the first is formed by adding the next integer to the previous term.


Recognizing such patterns is the first step in deriving a recursive formula. This helps transform a set of numbers into a well-defined sequence, making it easy to forecast future terms or verify given ones. By following the observed pattern, it's easy to build sequences in a logical, predictable fashion.
Arithmetic Sequences
Though the sequence \(1, 3, 6, 10, 15, \ldots\) is not an arithmetic sequence, understanding the basics of arithmetic sequences can provide a helpful comparison:

  • Arithmetic Sequences: Have a constant difference between consecutive terms. For instance, in the sequence \(2, 5, 8, 11, \ldots\), the common difference is \(3\).
  • Our Sequence: Has an increasing difference of consecutive integers, not a constant one \(2, 3, 4, \ldots\).


These differences highlight notable distinctions in progression types. While arithmetic sequences advance steadily at the same pace, sequences like ours develop more rapidly. This is because they accumulate increasingly larger increments鈥攎eaning the gap between terms grows. Understanding this distinction helps clarify why forming a recursive formula is crucial for sequences that don't fit the mold of simpler types like arithmetic sequences.

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Most popular questions from this chapter

Determine whether the series converges. $$\sum_{n=0}^{\infty} e^{-n}$$

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